with \(\mathbf {A}\)  and \(\mathbf {B}\)  the coefficients of this
polynomial. There are therefore a total of \((n+1)(n+2)\)  coefficients
to determine. Using a classical Levenberg-Marquardt
method[1]}, [2]}, the
position of the reference stars can be fitted to their known \((X_G,Y_G)\)  positions
on the sky and therefore determine the values of these coefficients
\(\mathbf {A}\)  and \(\mathbf {B}\) . The right part of
Fig. REF  shows in green color a grid of
\(13\times 13\)  stars around the telescope field center and in red color
the barycenters of the spot diagrams of Zemax ray tracing on the
detector. The positions in magenta are the result of the fitting of a
polynomial with order \(n=7\)  and 72 unknowns. In panel (c), the
distance between the detector position and the true position can be of
the order of several hundreds of pixels, whereas in panel (d) the
distance of the fitted and true position is only a small fraction of a
pixel (\(\simeq 10^{-5}\)  pixel).
